The author makes two statements I'm not at all clear about...can anyone explain??

In General Relativity, the change of the energy of a freely
moving photon should be the solution to the scalar equation
of isotropic geodesics, which is also known as the equation
of energy and manifests the work produced on the photon
being moved along the path.

What work is being done on a freely moving photon?

...the linear velocity .....of the rotation of space (due to the non-holonomity of it),

What does rotating space mean?....I thought there was rather convincing evidence our universe is not rotating....

I'll try on this one;
Just like dropping an object from a 100Km tower, the speed at first is slow,
After falling 100Km it will have much more Kinetic Energy.
The fall contributed “work” to boost that energy though acceleration to a higher speed.

Consider the photon that follows the same path; already moving at "c" with some energy.
Shouldn’t it also accelerate to a higher “speed” and energy?
Looking at the photon at the bottom we see it follows the rule of no change in speed, it is still only going at “c”.
However on closer inspection we also notice it has changed color to bluer!
That is a gain in Energy to a higher frequency;
the same “fall” has contributed the same amount of “work” to boost the energy of the photon.